Can Hamiltonians be boundary observables in parametrized field theories?

It has been argued that holography in gravitational theories is related to the existence of a particularly useful Gauss Law that allows energy to be measured at the boundary. This work investigates the extent to which such Gauss Laws follow from diffeomorphism invariance. We study parametrized field theories, which are a class of diffeomorphism-invariant theories without gravity. We find that the Hamiltonian for non-gravitational parametrized field theories vanishes on shell even in the presence of a boundary and under a variety of boundary conditions. We conclude that such theories have no useful Gauss Law, consistent with the absence of holography.